physics/9803042

IMSc/98/03/12 ∗ † ‡

THE DIRAC EQUATION APPROACH TOSPIN-1

2PARTICLE BEAM OPTICS

R. JAGANNATHANThe Institute of Mathematical Sciences

4th Cross Road, Central Institutes of Technology CampusTharamani, Chennai (Madras), Tamilnadu - 600 113, INDIA

E-mail: jagan@imsc.ernet.inhttp://www.imsc.ernet.in/∼jagan

The traditional approach to accelerator optics, based mainly on clas-sical mechanics, is working excellently from the practical point ofview. However, from the point of view of curiosity, as well as witha view to explore quantitatively the consequences of possible smallquantum corrections to the classical theory, a quantum mechanicalformalism of accelerator optics for the Dirac particle is being devel-oped recently. Here, the essential features of such a quantum beamoptical formalism for any spin-1

2 particle are reviewed. It seems thatthe quantum corrections, particularly those due to the Heisenberguncertainty, could be important in understanding the nonlinear dy-namics of accelerator beams.

∗Presented at the Workshop on “Quantum Aspects of Beam Physics”, 15th ICFA (Inter-national Committee for Future Accelerators) Advanced Beam Dynamics Workshop, January4 - 9, 1998, Monterey, California, U.S.A. To appear in the Proceedings of the Workshop, Ed.Pisin Chen (World Scientific, Singapore, 1998).†Keywords: Beam physics, Beam optics, Accelerator optics, Spin- 1

2particle, Anomalous

magnetic moment, Quantum mechanics, Dirac equation, Foldy-Wouthuysen transformation,Polarization, Thomas-Bargmann-Michel-Telegdi equation, Magnetic quadrupole lenses, Stern-Gerlach kicks, Nonlinear dynamics, Quantum corrections to the classical theory.‡PACS: 29.20.-c (Cyclic accelerators and storage rings), 29.27.-a (Beams in particle acceler-

ators), 29.27.Hj (Polarized beams), 41.75.-i (Charged-particle beams), 41.75.Ht (Relativisticelectron and positron beams), 41.85.-p (Beam optics), 41.85.Ja (Beam transport), 41.85.Lc(Beam focusing and bending magnets).

1 Introduction

Why should one bother about a quantum mechanical treatment of acceleratoroptics when the classical treatment works so well? This is a natural questionindeed. There is no prima facie reason at all to believe that a quantum me-chanical treatment would be necessary to deal with any aspect of acceleratoroptics design. As has been rightly pointed out [1], primary effects in conven-tional accelerators are essentially classical since the de Broglie wavelength of thehigh energy beam particle is much too small compared to the typical aperturesand the energy radiated is typically low and of long wavelength. However, it isbeing slowly recognized that quantum effects are still important due to demandson high precision in accelerator performance and ever increasing demands forhigher beam energy, luminosity and brightness [1]. Also, there is a growingfeeling now that a complete picture of spin polarization can only be obtainedon the basis of coupled spin↔ orbit phase-space transport equations (with andwithout radiation) and to include all the subtleties of radiation one has to beginwith quantum mechanics since classical white noise models simply would notsuffice for all situations [2]. I like to add:

• After all, accelerator beam is a quantum mechanical system and one maybe curious to know how its classical behavior can be understood, in detail,from a quantum mechanical formalism based on the appropriate relativis-tic wave equation.

• As has been revealed by recent studies [3] the passage from quantum the-ory to classical theory is not a simple affair, particularly when the systemis a complicated nonlinear dynamical system with regular and chaotic re-gions in its phase-space. Since accelerator beams are such systems [4]it is time that the quantum mechanics of accelerator optics is looked atseriously.

Essentially from the point of view of curiosity, the axially symmetric mag-netic lens was first studied [5] based completely on the Dirac equation. Laterworks [6, 7] led to further insights into the quantum mechanics of spin- 1

2 particlebeam optics. Quantum mechanics of the Klein-Gordon (spin-0) and nonrela-tivistic Schrodinger charged-particle beams were also studied [7, 8]. These worksdealt essentially with aspects of ion optics and electron optical imaging (for anexcellent survey of scalar electron wave optics see the third volume of the en-cyclopaedic three-volume text book of Hawkes & Kasper [9]; this contains alsoreferences to earlier works on the use of the Dirac equation in electron waveoptics problems like diffraction, to take into account the spinor nature of theelectron).

In the context of accelerator physics also, like in the case of electron andion beam optical device technologies, the practice of design of beam opticalelements is based mainly on classical physics. As is well known, various aspects

2

of accelerator beam dynamics like orbital motion, spin evolution and beampolarization, radiation and quantum fluctuations of trajectories, are analyzedpiecewise using classical, semiclassical, or quantum theories, or a mixture ofthem, depending on the situation treated. Quantum mechanical implicationsfor low energy polarized (anti)proton beams in a spin-splitter device, using thetransverse Stern-Gerlach (SG) kicks, have been analyzed [10] on the basis ofnonrelativistic Schrodinger equation.

To obtain the coupled spin↔ phase-space transport equations, for the spin- 12

particle, one needs an appropriate quantum Hamiltonian. Such a Hamiltonianwas stated, as following from the systematic Foldy-Wouthuysen (FW) trans-formation technique [11], by Derbenev and Kondratenko [12] in 1973 as thestarting point of their radiation calculations but no explicit construction wasgiven (such a Hamiltonian can also be justified [13] using the Pauli reductionof the Dirac theory). The Derbenev-Kondratenko (DK) Hamiltonian has beenused [14] to construct a completely classical approach to beam optics, includingspin components as classical variables. Now, a detailed derivation of the DKHamiltonian has been given [2] and a completely quantum mechanical formalismis being developed [2] in terms of the ‘machine coordinates’ and ‘observables’,suitable for treating the physics of spin- 1

2 polarized beams from the point ofview of machine design.

Independent of the DK formalism, recently [15, 16] we have made a beginningin the application of the formalism of the Dirac spinor beam optics, developedearlier ([5]-[7]) mostly with reference to electron microscopy, to accelerator opticsto understand in a unified way the orbital motion, SG kicks, and the Thomas-Bargmann-Michel-Telegdi (TBMT) spin evolution. Here, I present the essentialfeatures of our work, done so far, on the quantum beam optical approach toaccelerator optics of spin-1

2 particles based on the Dirac equation.

2 Quantum beam optics of the Dirac particle

Our formalism of quantum beam optics of the Dirac particle, in the context of ac-celerators, is only in the beginning stages of development. So, naturally there areseveral simplifying assumptions: We deal only with the single particle dynam-ics, based on the single particle interpretation of the Dirac equation, ignoring allthe inter-particle interactions and statistical aspects. Only monoenergetic beamis considered. The treatment is at the level of paraxial approximation, so far,though the general framework of the theory is such that extension to the caseof nonparaxial (nonlinear) systems is straightforward. Only time-independentmagnetic optical elements with straight axis are considered. Electromagneticfield is treated as classical. And, radiation is ignored.

Thus, we are dealing with elastic scattering of the particles of a monoen-ergetic beam by an optical element with a straight axis along the z-directionand comprising a static magnetic field B = curlA. Hence, the 4-component

3

spinor wavefunction of the beam particle can be assumed to be of the formΨ(r, t) = ψ(r) exp(−iEt/h), where E is the total (positive) energy of the parti-cle of mass m, charge q, and anomalous magnetic moment µa. The spatial partof the wavefunction ψ(r) has to obey the time-independent Dirac equation

Hψ(r) = Eψ(r) , (1)

where the Hamiltonian H, including the Pauli term, is given by

H = βmc2 + cα · π − µaβΣ ·B ,

β =

(1l 0000 −1l

), α =

(00 σσ 00

), Σ =

(σ 0000 σ

),

1l =

(1 00 1

), 00 =

(0 00 0

),

σx =

(0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

), (2)

with π = p − qA = −ih∇− qA. Let p be the design momentum of the beamparticle along the +z-direction so that E = +

√m2c4 + c2p2. The beam is

assumed to be paraxial: |p⊥| |p| = p and pz > 0.Since we are interested in studying the propagation of the beam along the

+z-direction we would like to rewrite Eq. (1) as

ih∂

∂zψ(r⊥; z) = Hψ(r⊥; z) . (3)

So, we multiply Eq. (1) from left by αz/c and rearrange the terms to get

H = −pβχαz − qAz1l + αzα⊥ · π⊥ + (µa/c)βαzΣ ·B ,

χ =

(ξ1l 0000 −ξ−11l

), 1l =

(1l 0000 1l

), ξ =

√E +mc2

E −mc2. (4)

Note that the matrix βχαz, coefficient of −p in H, is diagonalized as follows:

M(βχαz)M−1 = β , M =

1√

2(1l + χαz) . (5)

Hence, let us defineψ′ = Mψ . (6)

This turns Eq. (3) into

ih∂

∂zψ′ = H′ψ′ , H′ = MHM−1 = −pβ + E +O , (7)

4

with the ‘even’ operator E and the ‘odd’ operator O given, respectively, by thediagonal and off-diagonal parts of

E +O =

[−qAz1l− (µa/2c)×(ξ + ξ−1

)σ⊥ ·B⊥

+(ξ − ξ−1

)σzBz

]ξ [σ⊥ · π⊥ − (µa/2c)×i(ξ − ξ−1

)(Bxσy − Byσx)−(ξ + ξ−1

)Bz1l

]−ξ−1 [σ⊥ · π⊥ + (µa/2c)×i(ξ − ξ−1

)(Bxσy −Byσx)+(ξ + ξ−1

)Bz1l

][−qAz1l− (µa/2c)×(ξ + ξ−1

)σ⊥ ·B⊥

−(ξ − ξ−1

)σzBz

]

. (8)

The significance of the transformation in Eq. (6) is that for a paraxial beampropagating in the forward (+z) direction ψ′ is such that its lower pair of com-ponents are very small compared to the upper pair of components, exactly likefor a positive energy nonrelativistic Dirac spinor. Note the perfect analogy:nonrelativistic case → paraxial, positive energy → forward propagation, andmc2 → −p in Eq. (2) and Eq. (7) respectively.

Let us recall that the FW transformation technique [11, 17] is the mostsystematic way of analyzing the standard Dirac equation as a sum of the non-relativistic part and a series of relativistic correction terms. So, the applicationof an analogous technique to Eq. (7) should help us analyze it as a sum of theparaxial part and a series of nonparaxial correction terms. To this end, wedefine the FW-like transformation

ψ1 = S1ψ′ , S1 = exp (−βO/2p) . (9)

The resulting equation for ψ1 is

ih∂

∂zψ1 = H1ψ1 , H1 = S1H

′S−11 − ihS1

∂

∂z

S−1

1

= −pβ + E1 +O1 ,

E1 = E −1

2pβO2 + · · · , O1 = −

1

2pβ

[O, E ] + ih

∂

∂zO

+ · · · . (10)

A series of such transformations successively with the same type of recipe asin Eq. (9) eliminates the odd parts from H′ up to any desired order in 1/p.It should also be mentioned that these FW-like transformations preserve theproperty of ψ′ that its upper pair of components are large compared to thelower pair of components. We shall stop with the above first step which wouldcorrespond to the paraxial, or the first order, approximation.

Since the lower pair of components of ψ1 are almost vanishing compared toits upper pair of components and the odd part of H1 is negligible compared toits even part, up to the first order approximation we are considering, we caneffectively introduce a two-component spinor formalism based on the represen-tation of Eq. (10). Naming the two-component spinor comprising the upper

5

pair of components of ψ1 as ψ′′ and calling the 2× 2 11-block element of H1 asH′′ it is clear from Eq. (10) that we can write

ih∂

∂zψ′′ = H′′ψ′′ ,

H′′ ≈

(−p− qAz +

1

2pπ2⊥

)−

1

p(q + ε)BzSz + γεB⊥ · S⊥ ,

with π2⊥ = π2

x + π2y , ε = 2mµa/h , γ = E/mc2 , S = hσ/2. (11)

Up to now, all observables, field components, etc., are defined with referenceto the laboratory frame. But, as is usual in accelerator physics, we have todefine spin with reference to the instantaneous rest frame of the particle whilekeeping the other observables, field components, etc., defined with reference tothe laboratory frame. For this, we have to transform ψ′′ further to an ‘accel-erator optics representation’, say, ψA = TAψ

′′. The choice of TA is dictated bythe following consideration. Let the operator O correspond to an observable Oin the Dirac representation (Eq. (2) or Eq. (3)). The operator corresponding toO in the representation of Eq. (11) can be taken to be given by

O′′ = the hermitian part of the 11−block element of(S1MOM−1S−1

1

).

(12)The corresponding operator in the accelerator optics representation will be

OA = the hermitian part of(TAO

′′T−1A

). (13)

The operator

S(R) =1

2h

(σ − c2(σ·ππ+πσ·π)

2E(E+mc2)cπE

cπE −σ + c2(σ·ππ+πσ·π)

2E(E+mc2)

). (14)

corresponds to the rest-frame spin in the Dirac representation [18]. We demandthat the components of the rest-frame spin operator in the accelerator optics

representation be simply the Pauli spin matrices, i.e., S(R)A ≈ hσ/2, up to the

first order (paraxial) approximation. This demand leads to the choice

ψA = TAψ′′ , TA = exp −i (πxσy − πyσx) /2p . (15)

Now, finally, with the transformation given by Eq. (15), the desired basic equa-tion of the quantum beam optics of the Dirac particle becomes, up to paraxialapproximation,

ih∂

∂zψA = HAψA , HA ≈

(−p− qAz +

1

2pπ2⊥

)+γm

pΩ · S ,

with Ω = −1

γm

qB + ε

(B‖ + γB⊥

). (16)

6

Here,B‖ andB⊥ are the components ofB in the +z-direction (the predominantdirection of motion of the beam particles) and perpendicular to it, unlike inthe usual TBMT vector Ω in which the components B‖ and B⊥ are definedwith respect to the direction of the instantaneous velocity of the particle. Thequantum beam optical Hamiltonian HA is the beam optical version of the DKHamiltonian in the paraxial approximation. To get the higher order corrections,in terms of π⊥/p and h, we have to go beyond the first FW-like transformation.

It must be noted that while the exact quantum beam optical Dirac Hamilto-nianH (see Eq. (3)) is nonhermitian the nonunitary FW-like transformation hasprojected out a hermitianHA (see Eq. (16)). Thus, for the particle that survivesthe transport through the system, without getting scattered far away, ψA hasunitary evolution along the z-axis. Hence, we can normalize the two-componentψA, at any z, as 〈ψA(z) |ψA(z)〉 =

∫ ∫dxdy ψ†AψA = 1. This normalization will

be conserved along the optic (z) axis. Then, for any observable O representedby a hermitian operator OA, in the accelerator optics representation (Eq. (16)),we can define the average at the transverse plane at any z as

〈O〉(z) = 〈ψA(z)| OA |ψA(z)〉 =

∫ ∫dxdy ψ†AOAψA . (17)

Now, studying the z-evolution of 〈O〉(z) is straightforward. Integration ofEq. (16) gives

ψA(z′) = U(z′, z)ψA(z) , (18)

where the unitary z-evolution operator U(z′, z) can be obtained by the standardquantum mechanical methods. Thus, the relations

〈O〉(z′) = 〈ψA(z)| U(z′, z)†OAU(z′, z) |ψA(z)〉 , (19)

for the relevant set of observables, give the transfer maps for the quantumaverages (or their classical values a la Ehrenfest) from the plane at z to theplane at z′. In the classical limit this relation (Eq. (19)) becomes the basisfor the Lie algebraic approach to classical beam optics [19]. The Lie algebraicapproach has been studied [20] in the context of spin transfer map also usingthe classical formalism.

The main problem of accelerator optics is to know the transfer maps forthe quantum averages of the components of position, momentum, and the rest-frame spin between transverse planes containing the optical elements. We havealready seen that the rest-frame spin is represented in the accelerator opticsrepresentation by the Pauli spin matrices. Let us take that the observed posi-tion of the Dirac particle corresponds to the mean position operator of the FWtheory [11] or what is same as the Newton-Wigner position operator [21]. Then,one can show, using Eq. (12) and Eq. (13), that in the accelerator optics repre-sentation the transverse position operator is given by r⊥ up to the first orderapproximation (details are given elsewhere [22]). For the transverse momentumin free space the operator is p⊥ in the accelerator optics representation.

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3 An example: the normal magnetic quadru-pole lens

For an ideal normal magnetic quadrupole lens of length L comprising the fieldB = (Gy,Gx, 0), corresponding to A =

(0, 0, 1

2G(y2 − x2

)), and bounded by

transverse planes at zi and zf = zi+L, the quantum accelerator optical Hamil-tonian (Eq. (16)) becomes

HA =

−p+ 1

2p p2⊥ , for z < zi and z > zf ,

−p+ 12qG

(x2 − y2

)+ 1

2p p2⊥ −

(q+γε)Gh2p (yσx + xσy) ,

for zi ≤ z ≤ zf .

(20)

Note that the choice of A in a different gauge will not affect the average valuesdefined by Eq. (17). Now, using the formalism of the previous section, it isstraightforward to find the desired transfer maps in this case (details are foundelsewhere [15, 22]). The results are: with η = (q + γε)GLh/2p2, K =

√qG/p,

λ = h/p, 〈 〉i = 〈 〉(zi), and 〈 〉f = 〈 〉(zf ),(〈x〉f〈px〉f

)≈

(cosKL 1

pK sinKL

−pK sinKL cosKL

)×

((〈x〉i〈px〉i

)+ η

((cosKL− 1) 〈σy〉i /K

2L− (p sinKL) 〈σy〉i /KL

)),(

〈y〉f〈py〉f

)≈

(coshKL 1

pKsinhKL

pK sinhKL coshKL

)×

((〈y〉i〈py〉i

)+ η

(− (coshKL− 1) 〈σx〉i /K

2L− (p sinhKL) 〈σx〉i /KL

)),

〈Sx〉f ≈ 〈Sx〉i +4πη

λ

((sinKL

KL

)〈xSz〉i +

(cosKL− 1

K2Lp

)〈pxSz〉i

),

〈Sy〉f ≈ 〈Sy〉i −4πη

λ

((sinhKL

KL

)〈ySz〉i −

(coshKL− 1

K2Lp

)〈pySz〉i

),

〈Sz〉f ≈ 〈Sz〉i −4πη

λ

(sinKL

KL

)〈xSx〉i −

(sinhKL

KL

)〈ySy〉i

+

(cosKL− 1

K2Lp

)〈pxSx〉i +

(coshKL− 1

K2Lp

)〈pySy〉i

. (21)

Obviously we have obtained the well known classical transfer maps (matri-ces) for the transverse phase-space coordinates, and more, i.e., the transverseSG kicks [10, 23]. The longitudinal SG kick, which has been proposed [23] as abetter alternative to the transverse SG kicks for making a spin-splitter deviceto produce polarized (anti)proton beams, can also be understood [15, 22] usingthe present quantum beam optical formalism. The skew magnetic quadrupolecan also be analyzed [24] in the same way as here.

8

4 Conclusion

In summary, we have seen how one can obtain a fully quantum mechanical for-malism of the accelerator beam optics for a spin- 1

2 particle, with anomalousmagnetic moment, starting ab initio from the Dirac-Pauli equation. This for-malism leads naturally to a unified picture of orbital and spin dynamics takinginto account the effects of the Lorentz force, the SG force and the TBMT equa-tion for spin evolution. Only the lowest order (paraxial) approximation hasbeen considered in some detail, with an example. It is clear from the generaltheory, presented briefly here, that the approach is suitable for handling anymagnetic optical element with straight axis and computations can be carriedout to any order of accuracy by easily extending the order of approximation.It should be emphasized that the present formalism is valid for all values ofdesign momentum p from the nonrelativistic case to the ultrarelativistic case.The approximation scheme is based only on the fact that for a beam, consti-tuted by particles moving predominantly in one direction, the transverse kineticmomentum is very small compared to the longitudinal kinetic momentum.

We hope to address elsewhere [22] some of the issues related to the con-struction of a more general theory overcoming the limitations of the presentformalism. With reference to the inclusion of multiparticle dynamics within thepresent formalism, it might be profitable to be guided by the so-called thermalwave model which has been extensively developed [25] in recent years to accountfor the classical collective behavior of a charged-particle beam, by associatingwith the classical beam a quantum-like wavefunction obeying a Schrodinger-likeequation with the role of h played by the beam emittance ε.

To conclude, let me emphasize the significance of the quantum formalismfor beam optics. The following question has been raised: What are the usesof quantum formalisms in beam physics? [1] Of course, as we have seen above,we understand the quantum mechanics underlying the observed classical be-havior of the beam: when the h-dependent quantum corrections are worked outthrough the higher order FW-like transformations it is found that they are reallynegligible. In my opinion, quantum formalism of beam optics has more signif-icant uses. To see this, let me cite the following cases: (1) Recently there is arenewed interest [26] in the form of the force experienced by a spinning relativis-tic particle in an external electromagnetic field. Such studies are particularlyimportant in evaluating the possible mechanisms of spin-splitter devices [23].A thorough analysis based on general Poincare covariance, up to first order inspin, shows that classical spin-orbit systems can be characterized [27], at best,by five phenomenological parameters. This just points to the fact that spin be-ing essentially a quantum aspect one must eventually have a quantum formalismto understand really the high energy polarized accelerator beams. (2) A lookat Eq. (17) shows that the form of the transfer map for the quantum averagesof observables will differ from the form of the corresponding classical maps byterms of the type 〈f(O1, O2, . . . , )〉 − f(〈O1〉, 〈O2〉, . . . , ) 6= 0; the differences are

9

essentially due to the quantum uncertainties associated with ψA at the initial zi.For example, a term like x3 in a classical map will become in the correspondingquantum map 〈x3〉 = 〈x〉3 + 3〈x〉〈(x−〈x〉)2〉+ 〈(x−〈x〉)3〉 which need not van-ish even on the axis (where 〈x〉 = 0). It is thus clear that, essentially, quantummechanics modifies the coefficients of the various linear and nonlinear terms inthe classical map making them dependent on the quantum uncertainties asso-ciated with the wavefunction of the input beam; actually, even terms absent inthe classical map will be generated in this manner with coefficients dependenton the quantum uncertainties as a result of modifications of the other terms (arelated idea of fuzzy classical mechanics, or fuzzy quantum evolution equations,occurs in a different context [28]). This quantum effect could be significant inthe nonlinear dynamics of accelerator beams. This effect is relevant for spin dy-namics too. For example, it is seen in Eq. (21) that the spin transfer map is notlinear in its components, in principle, even in the lowest order approximation,since terms of the type, say, 〈xSz〉i, 〈pxSz〉i, etc., are not the same as 〈x〉i 〈Sz〉i,〈px〉i 〈Sz〉i, etc., respectively, in general.

Acknowledgments

In this First Book on Quantum Aspects of Beam Physics I would like to recordmy gratitude to Prof. E.C.G. Sudarshan for initiating my work on the topic ofDirac spinor beam optics. It is a pleasure to thank Prof. Pisin Chen, and theOrganizing Committee of QABP98, for sponsoring my participation in the his-toric Monterey meeting and for the hospitality I enjoyed during the conference.I am thankful to our Director, Prof. R. Ramachandran, for kind encouragementand my thanks are due to him, and to our Institute, for providing full financialsupport for my travel to participate in QABP98. I wish to thank Prof. SwapanChattopadhyay for bringing to my notice the literature on the application ofLie methods to spin dynamics.

References

[1] P. Chen, ICFA Beam Dynamics Newsletter 12, 46 (1996); P. Chen,“Overview of quantum beam physics”, Talk in QABP98.

[2] K. Heinemann and D.P. Barber, “The semiclassical Foldy-Wouthuysentransformation and the derivation of the Bloch equation for spin- 1

2 po-larized beams using Wigner functions”, Talk in QABP98 by D.P. Barber.

[3] See, e.g., G. Casati, Chaos 6, 391 (1996).

[4] See, e.g., H. Mais, “Some topics in beam dynamics of storage rings”, DESY96-119 (1996).

10

[5] R. Jagannathan, R. Simon, E.C.G. Sudarshan and N. Mukunda Phys. Lett.A 134,457 (1989); R. Jagannathan, in Dirac and Feynman: Pioneers inQuantum Mechanics, ed. R. Dutt and A.K. Ray (Wiley Eastern, New Delhi,1993).

[6] R. Jagannathan, Phys. Rev. A 42, 6674 (1990).

[7] R. Jagannathan and S.A. Khan, in Advances in Imaging and ElectronPhysics, Vol.97, ed. P.W. Hawkes (Academic Press, San Diego, 1996).

[8] S.A. Khan and R. Jagannathan, Phys. Rev. E 51, 2510 (1995).

[9] P.W. Hawkes and E. Kasper, Principles of Electron Optics - Volume 3:Wave Optics (Academic Press, San Diego, 1994).

[10] M. Conte and M. Pusterla, Il Nuovo Cimento A 103, 1087 (1990); M.Conte, Y. Onel, A. Penzo, A. Pisent, M. Pusterla and R. Rossmanith,“The spin-splitter concept”, INFN/TC-93/04.

[11] L.L. Foldy and S.A. Wouthuysen, Phys. Rev. 78, 29 (1950).

[12] Ya.S. Derbenev and A.M. Kondratenko, Soviet Phys. JETP 37, 968 (1973).

[13] J.D. Jackson, Rev. Mod. Phys. 48, 417 (1976).

[14] D.P. Barber, K. Heinemann and G. Ripken, Z. Phys. C 64, 117 (1994);D.P. Barber, K. Heinemann and G. Ripken, Z. Phys. C 64, 143 (1994).

[15] M. Conte, R. Jagannathan, S.A. Khan and M. Pusterla, Part. Accel. 56,99 (1996); S.A. Khan, Quantum Theory of Charged-Particle Beam Optics,Ph.D. Thesis (University of Madras, Chennai, 1997).

[16] R. Jagannathan and S.A. Khan, ICFA Beam Dynamics Newsletter 13, 21(1997).

[17] See also, e.g., J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics(McGraw-Hill, New York, 1964).

[18] See, e.g., A.A. Sokolov and I.M. Ternov, Radiation from Relativistic Elec-trons (American Inst. Phys., New York, 1986).

[19] See, e.g., A.J. Dragt, in Physics of High Energy Accelerators, AIP Conf.Proc. 87 (1982); A.J. Dragt, F. Neri, G. Rangarajan, D.R. Douglas, L.M.Healy and R.D. Ryne, Ann. Rev. Nucl. Part. Sci. 38, 455 (1988); E. For-est, M. Berz and J. Irwin, Part. Accel. 24, 91 (1989); E. Forest and K. Hi-rata, “A contemporary guide to beam dynamics”, KEK Report 92-12; A.J.Dragt, “Lie algebraic methods for ray and wave optics”, Talk in QABP98;and references therein.

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[20] K. Yokoya, “Calculation of the equilibrium polarization of stored electronbeams using Lie algebra”, Preprint KEK 86-90 (1986); Yu.I. Eidelmanand V.Ye. Yakimenko, “The spin motion calculation using Lie method incollider nonlinear magnetic field”, Preprint INP 90-127 (Novosibirsk, 1990);Yu.I. Eidelman and V.Ye. Yakimenko, Part. Accel. 45, 17 (1994).

[21] See, e.g., B. Thaller, The Dirac Equation (Springer, Berlin, 1992).

[22] M. Conte, R. Jagannathan, S.A. Khan and M. Pusterla, “A quantummechanical formalism for studying the transport of Dirac particle beamsthrough magnetic optical elements in accelerators”, in preparation.

[23] M. Conte, A. Penzo, and M. Pusterla, Il Nuovo Cimento A 108, 127 (1995);M. Pusterla, “Polarized beams and Stern-Gerlach forces in classical andquantum mechanics”, Talk in QABP98; and references therein.

[24] S.A. Khan, “Quantum theory of magnetic quadrupole lenses for spin- 12

particles”, Talk in QABP98.

[25] See R. Fedele and G. Miele, Il Nuovo Cimento D 13, 1527 (1991); and,e.g., R. Fedele, F. Gallucio, V.I. Man’ko and G. Miele, Phys. Lett. A 209,263 (1995); R. Fedele, “Quantum-like aspects of particle beam dynamics”,Talk in QABP98; and references therein.

[26] See, e.g., J. Anandan, Nature 387, 558 (1997); M. Chaichian. R.G. Felipeand D.L. Martinez, Phys. Lett. A 236, 188 (1997); J.P. Costella and B.H.J.McKellar, Int. J. Mod. Phys. A 9, 461 (1994); and references therein.

[27] K. Heinemann, “On Stern-Gerlach forces allowed by special relativ-ity and the special case of the classical spinning particle of Derbenev-Kondratenko”, DESY 96-229, 1996 (physics/9611001).

[28] J.L. Gruver, A.N. Proto and H.A. Cerdeira, “Fuzzy Classical Mechanics”,ICTP Preprint (1996).

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